Polypeptides and polynucleotides are natural programmable biopolymers that can self-assemble into complex tertiary structures. We describe a system analogous to designed DNA nanostructures in which protein coiled-coil (CC) dimers serve as building blocks for modular de novo design of polyhedral protein cages that efficiently self-assemble in vitro and in vivo. We produced and characterized )20 single-chain protein cages in three shapes—tetrahedron, four-sided pyramid, and triangular prism—with the largest containing )700 amino-acid residues and measuring 11 nm in diameter. Their stability and folding kinetics were similar to those of natural proteins. Solution small-angle X-ray scattering (SAXS), electron microscopy (EM), and biophysical analysis confirmed agreement of the expressed structures with the designs. We also demonstrated self-assembly of a tetrahedral structure in bacteria, mammalian cells, and mice without evidence of inflammation. A semi-automated computational design platform and a toolbox of CC building modules are provided to enable the design of protein cages in any polyhedral shape.
COBISS.SI-ID: 6266906
We analyze the data about works (papers, books) from the time period 1990-2010 that are collected in Zentralblatt MATH database. The data were converted into four 2-mode networks (works $\times$ authors, works $\times$ journals, works $\times$ keywords and works $\times$ mathematical subject classifications) and into a partition of works by publication year. The networks were analyzed using Pajek - a program for analysis and visualization of large networks. We explore the distributions of some properties of works and the collaborations among mathematicians. We also take a closer look at the characteristics of the field of graph theory as were realized with the publications.
COBISS.SI-ID: 17149273
Let $\Gamma$ be a connected $G$-arc-transitive graph, let $uv$ be an arc of $\Gamma$ and let $L$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. We study the problem of bounding $|G_{uv}|$ in terms of $L$ and the order of $\Gamma$.
COBISS.SI-ID: 16981849
A celebrated theorem of Tutte shows that the order of the vertex-stabiliser in a connected 3-valent arc-transitive graph does not exceed 48. It has long been known that no such bound exists in the broader context of connected 3-valent vertex-transitive graphs. In fact, several families of such graph in which the order of the vertex-stabiliser grows exponentially with the number of vertices were known. In this paper, we prove a surprising fact which shows that with the exception of these well understood families, in all other connected 3-valent vertex-transitive graphs the order of the vertex-stabiliser can be bounded above by a sublinear function of the order of the graph. The main result of this paper is that, if $\Gamma$ is a connected 4-valent $G$-arc-transitive graph and $v$ is a vertex of $\Gamma$, then either $\Gamma$ is one of a well understood infinite family of graphs, or $|G_v|\leq 2^43^6$ or $2|G_v|\log_2(|G_v|/2)\leq |V\Gamma|$ and that this last bound is tight. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs.
COBISS.SI-ID: 1537132228
Knots are some of the most remarkable topological features in nature. Self-assembly of knotted polymers without breaking or forming covalent bonds is challenging, as the chain needs to be threaded through previously formed loops in an exactly defined order. Here we describe principles to guide the folding of highly knotted single-chain DNA nanostructures as demonstrated on a nano-sized square pyramid. Folding of knots is encoded by the arrangement of modules of different stability based on derived topological and kinetic rules. Among DNA designs composed of the same modules and encoding the same topology, only the one with the folding pathway designed according to the "free-end" rule folds efficiently into the target structure. Besides high folding yield on slow annealing, this design also folds rapidly on temperature quenching and dilution from chemical denaturant. This strategy could be used to design folding of other knotted programmable polymers such as RNA or proteins.
COBISS.SI-ID: 5880858
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of $(2n + 1) \times (2n + 1)$ DASASMs is $\prod_{i=0}^n \frac{(3i)!}{(n+i)!}$, and a conjecture of Stroganov from 2008 that the ratio between the numbers of $(2n + 1) \times (2n + 1)$ DASASMs with central entry -1 and 1 is $n/(n + 1)$. Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.
COBISS.SI-ID: 18594137
On the back cover page: Configurations can be studied from a graph-theoretical viewpoint via the so-called Levi graphs and lie at the heart of graphs, groups, surfaces, and geometries, all of which are very active areas of mathematical exploration. In this self-contained textbook, algebraic graph theory is used to introduce groups; topological graph theory is used to explore surfaces; and geometric graph theory is implemented to analyze incidence geometries. After a preview of configurations in Chapter 1, a concise introduction to graph theory is presented in Chapter 2, followed by a geometric introduction to groups in Chapter 3. Maps and surfaces are combinatorially treated in Chapter 4. Chapter 5 introduces the concept of incidence structure through vertex colored graphs, and the combinatorial aspects of classical configurations are studied. Geometric aspects, some historical remarks, references, and applications of classical configurations appear in the last chapter. With over two hundred illustrations, challenging exercises at the end of each chapter, a comprehensive bibliography, and a set of open problems, Configurations from a Graphical Viewpoint is well suited for a graduate graph theory course, an advanced undergraduate seminar, or a self-contained reference for mathematicians and researchers.
COBISS.SI-ID: 16418137
Eff is a programming language based on the algebraic approach to computational effects, in which effects are viewed as algebraic operations and effect handlers as homomorphisms from free algebras. Eff supports first-class effects and handlers through which we may easily define new computational effects, seamlessly combine existing ones, and handle them in novel ways. We give a denotational semantics of Eff and discuss a prototype implementation based on it. Through examples we demonstrate how the standard effects are treated in Eff, and how Eff supports programming techniques that use various forms of delimited continuations, such as backtracking, breadth-first search, selection functionals, cooperative multi-threading, and others.
COBISS.SI-ID: 17192025
We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a twoparameter eigenvalue problem. The first determinantal representation is suitable for polynomials with scalar or matrix coeffcients and consists of matrices with asymptotic order $n^2/4$, where $n$ is the degree of the polynomial. The second representation is useful for scalar polynomials and has asymptotic order $n^2/6$. The resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10, as well as for polynomials with a small number of terms.
COBISS.SI-ID: 17644377
In the paper we show that the bibliographic data can be transformed into a collection of compatible networks. Using network multiplication different interesting derived networks can be obtained. In defining them an appropriate normalization should be considered. The proposed approach can be applied also to other collections of compatible networks. The networks obtained from the bibliographic data bases can be large (hundreds of thousands of vertices). Fortunately they are sparse and can be still processed relatively fast. We answer the question when the multiplication of sparse networks preserves sparseness. The proposed approaches are illustrated with analyses of collection of networks on the topic "social network" obtained from the Web of Science. The works with large number of co-authors add large complete subgraphs to standard collaboration network thus bluring the collaboration structure. We show that using an appropriate normalization their effect can be neutralized. Among other, we propose a measure of collaborativness of authors with respect to a given bibliography and show how to compute the network of citations between authors and identify citation communities.
COBISS.SI-ID: 16739929